Lemma 73.9.5 (0419)—The Stacks project

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Table of contents
Part 4: Algebraic Spaces
Chapter 73: Topologies on Algebraic Spaces
Section 73.9: Fpqc topology
Lemma 73.9.5 (cite)

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Lemma 73.9.5. Let $S$ be a scheme, and let $X$ be an algebraic space over $S$ . Suppose that $\mathcal{U} = \{ f_ i : X_ i \to X\} _{i \in I}$ is an fpqc covering of $X$ . Then there exists a refinement $\mathcal{V} = \{ g_ i : T_ i \to X\}$ of $\mathcal{U}$ which is an fpqc covering such that each $T_ i$ is a scheme.

Proof. Omitted. Hint: For each $i$ choose a scheme $T_ i$ and a surjective étale morphism $T_ i \to X_ i$ . Then check that $\{ T_ i \to X\}$ is an fpqc covering. $\square$

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